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Chapter 9: Q12. (page 336)

Discover and prove a theorem about two lines tangent to a circle at the endpoints of a diameter.

Short Answer

Expert verified

Lines are parallel.

Step by step solution

01

Step 1. Given information.

Let a circle with centre O and diameter CD. Let MN be the tangent at point C and PQ be the tangent at point D.

02

Step 2. Formula used.

Tangent at any point of circle is perpendicular to the radius through point of contact.

03

Step 3. Proof.

Consider the figure below,

Here, MNis tangent at point C and PQis tangent at point D. Therefore,

OC鈯�MNOD鈯�PQ

Therefore,

鈭�OCN=90掳鈭�ODQ=90掳

i.e., 鈭�OCM=90鈭�and鈭�ODP=90掳

For lines MNand PQand transversal AB, 鈭�DCM=鈭�CDQ,

both alternate angels are equal.

Therefore, lines are parallel.

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In exercises 2-7 find the measure of the arc.

role="math" localid="1649086294011" AC铃�

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